How do I show something is pathwise connected?

  • #1
I need to prove that X={(x,y):a<=x<=b, c<=y<=d}

I was thinking of using proof by contradiction.

Assume that X is not pathwise connected, then for a,b in X there is no continuous function that connects the two.

I can show that then the set is disconnected but not sure where to go after that.

Am I going about it the correct way?

Thanks!
 

Answers and Replies

  • #2
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,129
3,301
Why can't you just give the path that connects two points??

Given (x,y) and (u,v). What path can you take that connects (x,y) and (u,v)????
 
  • #3
if I take a straight line path from the points, it would be the distance between the two, so sqrt((x-u)^2 + (y-v)^2)
 
  • #4
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,129
3,301
I don't care about the distance. I want a function [itex][0,1]\rightarrow X[/itex]. A linear path is ok. How do you write that in function language?
 
  • #5
i am not sure what you mean by function language but I would want f(0) = (x,y) and f(1) = (u, v)
 
  • #6
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,129
3,301
Xhat is the equation of the line connecting your two points??
 
  • #7
i want to say something like

given f(0) = (x, y) and p in [0,1] then f(p) = ((y-v)/(x-u)) * p + (x, y)

but thats not correct bc i am switching between R and R^2

do i need to make a continuous function for each component?
 
  • #8
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,129
3,301
Yes, you need to go linearly drom x to u and from y to v.
 
  • #9
do I do it peicewise? like f(p) = (x,y) for p=0, (u, v) for p=1 and then this other functions that I cannot figure out how to get.
 
  • #10
HallsofIvy
Science Advisor
Homework Helper
41,847
969
I need to prove that X={(x,y):a<=x<=b, c<=y<=d}
You need to prove what about this set? That it is connected? Or that it is pathwise connected? A set can be "connected" without being "pathwise connected".

I was thinking of using proof by contradiction.

Assume that X is not pathwise connected, then for a,b in X there is no continuous function that connects the two.

I can show that then the set is disconnected but not sure where to go after that.

Am I going about it the correct way?

Thanks!
 
  • #11
Pathwise connected
 

Related Threads on How do I show something is pathwise connected?

Replies
1
Views
2K
Replies
5
Views
1K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
8
Views
9K
Replies
6
Views
6K
  • Last Post
Replies
9
Views
4K
Replies
2
Views
7K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
4K
  • Last Post
Replies
4
Views
3K
Top